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Is there a series representation for $\frac{1}{\log(x)}$?

for $x>0$ it is known that: $$\log(x)=2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}$$ Is there a series representation for $\frac{1}{\log(x)}$ in the following form? $$\frac{1}{\log(x)}=a_1(x)+a_2(x)+a_3(x)+\dots=\sum_{k=1}^\infty a_k(x)$$ Or, is there a way to transform the expression $$ \frac{1}{2\sum_{k=1}^\infty \frac{\frac{x-1}{x+1}^{2k-1}}{2k-1}}$$ into the aforementioned form?